The connection between the $PQ$ penny flip game and the dihedral groups

TL;DR

This paper explores the PQ penny flip game using group theory, revealing the structure of winning strategies within dihedral groups and extending the analysis to quantum moves and larger groups.

Contribution

It establishes a group-theoretic framework for the PQ penny flip game, identifying classes of winning strategies and extending results to all dihedral groups and quantum move sets.

Findings

Two classes of winning strategies for Q within D8.

Exact sequences of states guaranteeing Q's win with probability 1.

Extension of the game analysis to all D8n dihedral groups and quantum moves.

Abstract

This paper is inspired by the PQ penny flip game. It employs group-theoretic concepts to study the original game and also its possible extensions. We show that the PQ penny flip game can be associated with the dihedral group $D_{8}$. We prove that within $D_{8}$ there exist precisely two classes of winning strategies for Q. We establish that there are precisely two different sequences of states that can guaranteed Q's win with probability $1.0$. We also show that the game can be played in the all dihedral groups $D_{8 n}$, $n \geq 1$, with any significant change. We examine what happens when Q can draw his moves from the entire $U(2)$ and we conclude that again, there are exactly two classes of winning strategies for Q, each class containing now an infinite number of equivalent strategies, but all of them send the coin through the same sequence of states as before. Finally, we consider general extensions of the game with the quantum player having $U(2)$ at his disposal. We prove that for Q to surely win against Picard, he must make both the first and the last move.